EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3386-3398
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Inclusion Properties for Hohlov Operator to be in
Comprehensive Subfamilies of Analytic Functions

Tariq Al-Hawary1,3,∗, Mourad Oqla Massa’deh1, Ahlam Omar Fallatah2

1 Department of Applied Science, Ajloun College, Al Balqa Applied University,
Ajloun 26816. Jordan
2 Taibah University, College of Science, Department of Mathematics, Madinah,
Saudi Arabia
3 Jadara Research Center, Jadara University, Irbid 21110, Jordan

Abstract. In this paper, the subfamilies Cκ3
(κ1, κ2) and S∗κ3

(κ1, κ2) investigated through the
Hohlov operator. More specifically, a number of sufficient requirements are given in order for the
aforementioned functions subfamilies. Moreover, our results will imply several corollaries.

2020 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic function, Geometric functions, Gaussian function, Hohlov
operator

1. Preliminaries and Definitions

Geometric functions are a basic family of special functions in mathematics that have
complex mathematical features and linkages and are widely used in many different domains
as solutions to differential equations and recurrence relations. Gaussian hypergeometric
function is one of the most important in geometric functions theory, it is foundation is
found in the research of 17th-century mathematicians and astrophysicists. In terms of
measurement, instrumentation, and statistics, the Gaussian distribution is most likely
the most employed. Indeed, celestial sources frequently have a Gaussian nature in radio
astronomy [8].

Let Π be the family of all analytic and univalent functions of the form:

L(ς) = ς +
∞∑
τ=2

rτ ς
τ , |ς| < 1, (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5407

Email addresses: tariqamh@bau.edu.jo (T.Al −Hawary),
mourad.oqla@bau.edu.jo (M. O. Massa’deh), afallatah@taibahu.edu.sa (A. O. Fallatah)

https://www.ejpam.com 3386 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3387

which are normalized by L(0) = L′(0) − 1 = 0. Also, indicate by H the subfamily of Π
made up of functions of the form:

L(ς) = ς −
∞∑
τ=2

rτ ς
τ , rτ ≥ 0. (2)

For two functions L, ℓ ∈ Π, L given by (1) and ℓ(ς) = ς +
∞∑
τ=2

sτ ς
τ , we define the

convolution of L and ℓ by:

(L∗ℓ)(ς) = ς +
∞∑
τ=2

rτsτ ς
τ .

The subfamily ξ − ST meeting the requirements

ξ − ST =

{
L∈ Π : Re

(
ςL′(ς)

L(ς)

)
> ξ

∣∣∣∣ ςL′(ς)

L(ς)
− 1

∣∣∣∣ , (0 ≤ ξ < ∞, |ς| < 1)

}
and L(ς) ∈ ξ − UCV ⇔ ςL′(ς) ∈ ξ − ST , was presented by Kanas and Wísniowska [13]
and [14]. In particular, when ξ = 1, we obtain the comprehensive subfamilies of uniformly
convex functions UCV and parabolic starlike functions SP in U = { ς ∈ C : |ς| < 1}. Also,
when ξ = 0, we obtain the renowned subfamilies of convex functions CV and starlike
functions ST in U (see for details, [10]). Let (see [13] and [14])

Υ1 =: Υ1(ξ) =


8(cos−1 ξ)

2

π2(1−ξ2) for 0 ≤ ξ < 1
8
π2 for ξ = 1
π2

4
√
v(v+1)(ξ2−1)ψ2(v)

for ξ > 1,

(3)

where v ∈ (0, 1) is determined by ξ = cosh (πF ′(v)⧸4F (v)), F is the Legendre’s complete

Elliptic integral of the first kind F (v) =
1∫
0

dt
(1−t2)(1−v2t2) and F ′(v) = F

(√
1− v2

)
is the

complementary integral of F (v).
Let L∈ Π be of the form (2) and in the subfamily ξ−UCV, then the following inequal-

ities hold true [13]

|rτ | ≤
(Υ1(ξ))τ−1

τ !
, τ ∈ N− {1}. (4)

Also, if L∈ Π be of the form (2) in the subfamily ξ − ST , then [14]

|rτ | ≤
(Υ1(ξ))τ−1

(τ−1)!
, τ ∈ N− {1}. (5)

The Gaussian hypergeometric function G(ℶ1,ℶ2;ℶ3; ς) given by:

G(ℶ1,ℶ2;ℶ3; ς) =

∞∑
τ=0

(ℶ1)τ (ℶ2)τ
(ℶ3)τ (1)τ

ςτ , |ς| < 1



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3388

where ℶ1,ℶ2,ℶ3 ∈ C such that ℶ3 ̸= 0,−1,−2, · · · , (ℶ1)0 = 1 for ℶ1 ̸= 0 and for τ ∈ N,
(ℶ1)τ = ℶ1(ℶ1 + 1)(ℶ1 + 2) · · · (ℶ1 + τ − 1) is the Pochhammer symbol, and it represents
the solution of the homogenous differential equation

ς(1− ς)y′′(ς) + (ℶ3 − (ℶ1 + ℶ2+1)ς) y′(ς)− ℶ1ℶ2y(ς)=0

has several uses in a variety of fields, including continued fractions, quasi-conformal theory,
conformal mappings, and more. Using Gauss Summation theorem, it is possible to write

G(ℶ1,ℶ2;ℶ3; 1) =
∞∑
τ=0

(ℶ1)τ (ℶ2)τ
(ℶ3)τ (1)τ

=
Γ(ℶ3 − ℶ2 − ℶ1)Γ(ℶ3)

Γ(ℶ3 − ℶ1)Γ(ℶ3 − ℶ2)
, for Re(ℶ3 − ℶ2 − ℶ1) > 0.

(6)
For L∈ Π, we recall the operator HOℶ1,ℶ2,ℶ3(L) : Π → Π of Hohlov [11] defined as:

HOℶ1,ℶ2,ℶ3(L) ≡ HOℶ1,ℶ2,ℶ3(L)(ς) = ςG(ℶ1,ℶ2;ℶ3; ς) ∗ L(ς)

= ς +

∞∑
τ=2

(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

rτ ς
τ .

Remark 1. 1) If ℶ1 = 1, ℶ2 = α + 1, ℶ3 = α + 2 with Re(α) > −1, then the operator
HOℶ1,ℶ2,ℶ3(L) turns into HO1,α+1,α+2(L) Bernardi operator [7],

2) HO1,1,2(L) Alexander operator [3],
3) HO1,2,3(L) Libera operator [16].

We examine the following subfamilies of analytic functions investigated by Murugusun-
daramoorthy et al., [18] and Ali et al., [20], respectively.

Definition 1. For some κ1(0 ≤ κ1 < 1), κ2(κ2 ≥ 0) and κ3(0 ≤ κ3 ≤ 1). Let the subfamily
Cκ3(κ1, κ2) consists of functions in Π satisfying the inequality

Re

(
ςL′(ς) + ς2L′′(ς)

(1− κ3)ς + κ3ςL′(ς)
− κ1

)
> κ2

∣∣∣∣ ςL′(ς) + ς2L′′(ς)

(1− κ3)ς + κ3ςL′(ς)
− 1

∣∣∣∣ , |ς| < 1

and the subfamily S∗κ3(κ1, κ2) consists of functions in Π satisfying the inequality

Re

(
ςL′(ς)

(1− κ3)ς + κ3L(ς)
− κ1

)
> κ2

∣∣∣∣ ςL′(ς)

(1− κ3)ς + κ3L(ς)
− 1

∣∣∣∣ , |ς| < 1.

Example 1. [12, 19] For some κ1(0 ≤ κ1 < 1), κ2(κ2 ≥ 0), κ3 = 1 and L(ς) of the form
(1), let the subfamily C1(κ1, κ2) consists of functions in Π satisfying

Re

(
ςL′′(ς)

L′(ς)
+ 1− κ1

)
> κ2

∣∣∣∣ ςL′′(ς)

L′(ς)

∣∣∣∣ , |ς| < 1

and the subfamily S∗1(κ1, κ2) consists of functions in Π satisfying

Re

(
ςL′(ς)

L(ς)
− κ1

)
> κ2

∣∣∣∣ ςL′(ς)

L(ς)
− 1

∣∣∣∣ , |ς| < 1.



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3389

Example 2. [18] For some κ1(0 ≤ κ1 < 1), κ2(κ2 ≥ 0), κ3 = 0 and L(ς) of the form (1),
let the subfamily C0(κ1, κ2) consists of functions in Π satisfying

Re
((

ςL′(ς)
)′ − κ1

)
> κ2

∣∣∣(ςL′(ς)
)′ − 1

∣∣∣ , |ς| < 1

and the subfamily S∗0(κ1, κ2) consists of functions in Π satisfying

Re
(
L′(ς)− κ1

)
> κ2

∣∣L′(ς)− 1
∣∣ , |ς| < 1.

Example 3. [21] For some κ1(0 ≤ κ1 < 1), κ2 = 0, κ3 = 1 and L(ς) of the form (1), let
the subfamily C∗

1(κ1, 0) ≡ CV(κ1) consists of functions in Π satisfying

Re

(
ςL′′(ς)

L′(ς)
+ 1

)
> κ1, |ς| < 1

and the subfamily S∗1(κ1, 0) ≡ ST (κ1) consists of functions in Π satisfying

Re

(
ςL′(ς)

L(ς)

)
> κ1, |ς| < 1.

Both subfamilies CV (κ1) and ST (κ1) are well known subfamilies of convex and starlike
functions of order κ1, respectively. Moreover, if κ1 = κ2 = 0, κ3 = 1, we get the subfamilies
of convex functions CV and starlike functions ST , respectively (see [21]).

Lemma 1. [18] A function L ∈ Cκ3(κ1, κ2) if and only if

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)] |rτ | ≤ 1− κ1 (7)

and L ∈ S∗κ3(κ1, κ2) if and only if

∞∑
τ=2

[(τ(κ2 + 1)− κ3(κ1 + κ2)] |rτ | ≤ 1− κ1. (8)

The geometric characteristics of numerous types of special functions are covered in a
substantial body of literature ( see [1], [6], [9], [4], [11], [2]).

The current paper aims to create connections between Hohlov operator and geometric
function theory. By findings the relationships between different subfamilies of analytic
univalent functions. Inspired by numerous works, for example Ahmad et al. [17] deduced
sufficient conditions and some properties for Mittag-Leffler. Frasin et al. [5] deduced
necessary and sufficient conditions for Struve functions to be in some classes. Kasthuri et
al. [15] introduced a new class by Hohlov Operator and obtain some inclusion relations.
Murugusundaramoorthy et al. [18] introduced a class of starlike functions and subordi-
nation results for some classes of starlike functions. Swaminathan [22] introduced some
conditions for normalized Gaussian hypergeometric function.



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3390

2. Main Results

This section will provide sufficient conditions for Hohlov operator function to be in the
subfamilies Cκ3(κ1, κ2) and S∗κ3(κ1, κ2).

Theorem 1. If L ∈ ST and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ Cκ3(κ1, κ2)
if the following condition is satisfied:

(κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (|ℶ1|+ 2) (|ℶ2|+ 2)

ℶ3(ℶ3 + 1)(ℶ3 + 2)
G2,1(|ℶ1|+ 3, |ℶ2|+ 3;ℶ3 + 3,1)

+
(6(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2,1)

+
(7(κ2 + 1)− 3κ3(κ1 + κ2)) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1,1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G2,1 |ℶ1| , |ℶ2| ;ℶ3,1)− 1] ≤ 1− κ1.

Proof. By equation (7), to prove HOℶ1,ℶ2,ℶ3(L) ∈ Cκ3(κ1, κ2), it suffices to show that

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

rτ

∣∣∣∣ ≤ 1− κ1.

Since L ∈ ST we have |rτ | ≤ τ , from above equation we get

∞∑
τ=2

[(
τ3(κ2 + 1

)
− τ2κ3(κ1 + κ2)

] ∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣ ≤ 1− κ1.

Writing 
τ = (τ − 1) + 1,

τ2 = (τ − 1)(τ − 2) + 3(τ − 1) + 1,
τ3 = (τ − 1)(τ − 2)(τ − 3) + 6(τ − 1)(τ − 2) + 7(τ − 1) + 1

and use the relations

(ℶ1)τ = ℶ1(ℶ1 + 1)τ−1 and |(ℶ1)τ | ≤ (|ℶ1|)τ , (9)

we have

∞∑
τ=2

[(
τ3(κ2 + 1

)
− τ2κ3(κ1 + κ2)

] ∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣
= (κ2 + 1)

∞∑
τ=2

(τ − 1)(τ − 2)(τ − 3)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

+(6(κ2 + 1)− κ3(κ1 + κ2))

∞∑
τ=2

(τ − 1)(τ − 2)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3391

+(7(κ2 + 1)− 3κ3(κ1 + κ2))

∞∑
τ=2

(τ − 1)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

+(κ2 − κ3(κ1 + κ2) + 1)
∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

≤ (κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (|ℶ1|+ 2) (|ℶ2|+ 2)

ℶ3(ℶ3 + 1)(ℶ3 + 2)

∞∑
τ=4

(|ℶ1|+ 3)τ−4 (|ℶ2|+ 3)τ−4

(ℶ3 + 3)τ−4 (1)τ−4

+
(6(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)

∞∑
τ=3

(|ℶ1|+ 2)τ−3 (|ℶ2|+ 2)τ−3

(ℶ3 + 2)τ−3 (1)τ−3

+
(7(κ2 + 1)− 3κ3(κ1 + κ2)) |ℶ1ℶ2|

ℶ3

∞∑
τ=2

(|ℶ1|+ 1)τ−2 (|ℶ2|+ 1)τ−2

(ℶ3 + 1)τ−2 (1)τ−2

+(κ2 − κ3(κ1 + κ2) + 1)

∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

.

By Gauss Summation Theorem, we can write

∞∑
τ=2

[(
τ2(κ2 + 1

)
− τκ3(κ1 + κ2)

] ∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣
≤ (κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (|ℶ1|+ 2) (|ℶ2|+ 2)

ℶ3(ℶ3 + 1)(ℶ3 + 2)
G2,1(|ℶ1|+ 3, |ℶ2|+ 3;ℶ3 + 3,1)

+
(6(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2,1)

+
(7(κ2 + 1)− 3κ3(κ1 + κ2)) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1, 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3,1)− 1] . (10)

But the expression (10) is bounded above by 1− κ1, thus the proof is completed.

Theorem 2. If L ∈ CV and ℶ1,ℶ2 ∈ C− {0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ Cκ3(κ1, κ2)
if the following condition is satisfied:

(κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2,1)

+
(3(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1,1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3,1)− 1] ≤ 1− κ1.

Proof. Since L ∈ CV we have |rτ | ≤ 1 and by equation (8), to prove HOℶ1,ℶ2,ℶ3(L) ∈
Cκ3(κ1, κ2), it suffices to show that

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

rτ

∣∣∣∣



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3392

≤
∞∑
τ=2

[(
τ2(κ2 + 1

)
− τκ3(κ1 + κ2)

] ∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣ ≤ 1− κ1.

Writing τ= (τ−1) + 1, τ2 = (τ−1)(τ−2) + 3(τ−1) + 1 and use of (9), we get

∞∑
τ=2

[(
τ2(κ2 + 1

)
− τκ3(κ1 + κ2)

] ∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣
= (κ2 + 1)

∞∑
τ=2

(τ − 1)(τ − 2)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

+(3(κ2 + 1)− κ3(κ1 + κ2))
∞∑
τ=2

(τ − 1)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

+(κ2 − κ3(κ1 + κ2) + 1)
∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

≤ (κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)

∞∑
τ=3

(|ℶ1|+ 2)τ−3 (|ℶ2|+ 2)τ−3

(ℶ3 + 2)τ−3 (1)τ−3

+
(3(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2|

ℶ3

∞∑
τ=2

(|ℶ1|+ 1)τ−2 (|ℶ2|+ 1)τ−2

(ℶ3 + 1)τ−2 (1)τ−2

+(κ2 − κ3(κ1 + κ2) + 1)

∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

.

By Gauss Summation Theorem, we have

∞∑
τ=2

[(
τ2(κ2 + 1

)
− τκ3(κ1 + κ2)

] ∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣
≤ (κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2, 1)

+
(3(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1, 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3, 1)− 1] . (11)

But the expression (11) is bounded above by 1− κ1, thus the proof is completed.

Theorem 3. Let Υ1 given by (3) and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R. If L ∈ ξ−UCV for some
ξ(0 ≤ ξ < ∞) and satisfies the inequality

(κ2 + 1) |ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] ≤ 1− κ1,

then HOℶ1,ℶ2,ℶ3(L) ∈ Cκ3(κ1, κ2).



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3393

Proof. To prove HOℶ1,ℶ2,ℶ3(L) ∈ Cκ3(κ1, κ2), it suffices to show that

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

rτ

∣∣∣∣ ≤ 1− κ1.

Applying the inequality (4) and use of (9), and then write τ = (τ − 1) + 1, we get

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

rτ

∣∣∣∣
≤

∞∑
τ=2

[(τ(κ2 + 1)− κ3(κ1 + κ2)]
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

(Υ1)τ−1

(1)τ−1

≤ (κ2 + 1)
∞∑
τ=2

(τ − 1)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

+(κ2 − κ3(κ1 + κ2) + 1)
∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

≤ (κ2 + 1) |ℶ1ℶ2|Υ1

ℶ3(1)1

∞∑
τ=2

(|ℶ1|+ 1)τ−2 (|ℶ2|+ 1)τ−2 (Υ1 + 1)τ−2

(ℶ3 + 1)τ−2 (1)τ−2(2)τ−2

+(κ2 − κ3(κ1 + κ2) + 1)

∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1 (Υ1)τ−1

(ℶ3)τ−1 (1)τ−1(1)τ−1

.

Using Gauss Summation Theorem, we can write

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣
≤ (κ2 + 1) |ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] . (12)

But the expression (12) is bounded above by 1− κ1, thus the proof is completed.

Theorem 4. Let Υ1 given by (3) and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R. If L ∈ ξ−ST for some
ξ(0 ≤ ξ < ∞) and satisfies the inequality

(κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (Υ1 + 1)Υ1

ℶ3(ℶ3+1)(2)1
G3,2(|ℶ1|+ 2, |ℶ2|+ 2,Υ1 + 2;ℶ3 + 2, 3; 1)

+
(3(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] ≤ 1− κ1,

then HOℶ1,ℶ2,ℶ3(L) ∈ Cκ3(κ1, κ2).



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3394

Proof. To prove HOℶ1,ℶ2,ℶ3(L) ∈ Cκ3(κ1, κ2), it suffices to show that

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

rτ

∣∣∣∣ ≤ 1− κ1.

Applying the inequality (5) and use of (9), and then write τ = (τ − 1) + 1, τ2 =
(τ−1)(τ−2) + 3(τ−1) + 1, we get

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

rτ

∣∣∣∣
≤

∞∑
τ=2

[(
τ2(κ2 + 1

)
− τκ3(κ1 + κ2)

] (|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

(Υ1)τ−1

(1)τ−1

≤ (κ2 + 1)
∞∑
τ=2

(τ − 1)(τ − 2)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

(Υ1)τ−1

(1)τ−1

+(3(κ2 + 1)− κ3(κ1 + κ2))
∞∑
τ=2

(τ − 1)
(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

(Υ1)τ−1

(1)τ−1

+(κ2 − κ3(κ1 + κ2) + 1)

∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1

(ℶ3)τ−1 (1)τ−1

(Υ1)τ−1

(1)τ−1

≤ (κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (Υ1 + 1)Υ1

ℶ3(ℶ3+1)(2)1

∞∑
τ=3

(|ℶ1|+ 2)τ−3 (|ℶ2|+ 2)τ−3 (Υ1 + 2)τ−3

(ℶ3 + 2)τ−3 (1)τ−3(3)τ−3

(3(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2|Υ1

ℶ3(1)1

∞∑
τ=2

(|ℶ1|+ 1)τ−2 (|ℶ2|+ 1)τ−2 (Υ1 + 1)τ−2

(ℶ3 + 1)τ−2 (1)τ−2(2)τ−2

+(κ2 − κ3(κ1 + κ2) + 1)
∞∑
τ=2

(|ℶ1|)τ−1 (|ℶ2|)τ−1 (Υ1)τ−1

(ℶ3)τ−1 (1)τ−1(1)τ−1

.

By Gauss Summation Theorem, we can write

∞∑
τ=2

τ [(τ(κ2 + 1)− κ3(κ1 + κ2)]

∣∣∣∣(ℶ1)τ−1(ℶ2)τ−1

(ℶ3)τ−1(1)τ−1

∣∣∣∣
≤ (κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (Υ1 + 1)Υ1

ℶ3(ℶ3+1)(2)1
G3,2(|ℶ1|+ 2, |ℶ2|+ 2,Υ1 + 2;ℶ3 + 2, 3; 1)

+
(3(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] . (13)

But the expression (13) is bounded above by 1− κ1, thus the proof is completed.

Using the same proceeding used to prove Theorem 1, Theorem 2 and Theorem 4, we
get the following Theorems for subfamily S∗κ3(κ1, κ2), respictivlly.



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3395

Theorem 5. If L ∈ ST and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ S∗κ3(κ1, κ2)
if the following condition is satisfied:

(κ2 + 1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3+1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2, 1)

+
(3(κ2 + 1)− κ3(κ1 + κ2)) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1, 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3, 1)− 1] ≤ 1− κ1.

Theorem 6. If L ∈ CV and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ S∗κ3(κ1, κ2)
if the following condition is satisfied:

(κ2 + 1) |ℶ1ℶ2|
ℶ3

G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1, 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3, 1)− 1] ≤ 1− κ1.

Theorem 7. Let Υ1 given by (3) and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R. If L ∈ ξ−ST for some
ξ(0 ≤ ξ < ∞) and satisfies the inequality

(κ2 + 1) |ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (κ2 − κ3(κ1 + κ2) + 1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] ≤ 1− κ1,

then HOℶ1,ℶ2,ℶ3(L) ∈ S∗κ3(κ1, κ2).

3. Some Corollaries

In this section, by suitable choices for parameters ℶ1, ℶ2 and ℶ3, we can conclude many
subresults from our main results related to Bernardi operator HO1,α+1,α+2(L), Alexander
operator HO1,1,2(L) and Libera operator HO1,2,3(L). Also, by suitable choices for param-
eters κ1, κ2 and κ3, we have many subresults. For example, if we set κ2 = 0 and κ3 = 1
in our main results, we get the following corollaries for subfamilies CV(κ1) and ST (κ1).

Corollary 1. If L ∈ ST and ℶ1,ℶ2 ∈ C− {0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ CV(κ1) if

|ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (|ℶ1|+ 2) (|ℶ2|+ 2)

ℶ3(ℶ3 + 1)(ℶ3 + 2)
G2,1(|ℶ1|+ 3, |ℶ2|+ 3;ℶ3 + 3,1)

+
(6− κ1) |ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2,1)

+
(7− 3κ1) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1,1)

+ (1−κ1) [G2,1 |ℶ1| , |ℶ2| ;ℶ3,1)− 1] ≤ 1− κ1.



T. Al-Hawary, M. O. Massa’deh, A. O Fallatah / Eur. J. Pure Appl. Math, 17 (4) (2024), 3386-3398 3396

Corollary 2. If L ∈ CV and ℶ1,ℶ2 ∈ C− {0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ CV(κ1) if
|ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3 + 1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2,1)

+
(3− κ1) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1,1)

+ (1−κ1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3,1)− 1] ≤ 1− κ1.

Corollary 3. Let Υ1 given by (3) and ℶ1,ℶ2 ∈ C − {0}, ℶ3 ∈ R. If L ∈ ξ − UCV for
some ξ(0 ≤ ξ < ∞) and satisfies

|ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (1−κ1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] ≤ 1− κ1,

then HOℶ1,ℶ2,ℶ3(L) ∈ K(κ1).

Corollary 4. Let Υ1 given by (3) and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R. If L ∈ ξ−ST for some
ξ(0 ≤ ξ < ∞) and satisfies

|ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1) (Υ1 + 1)Υ1

ℶ3(ℶ3+1)(2)1
G3,2(|ℶ1|+ 2, |ℶ2|+ 2,Υ1 + 2;ℶ3 + 2, 3; 1)

+
(3− κ1) |ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (1− κ1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] ≤ 1− κ1,

then HOℶ1,ℶ2,ℶ3(L) ∈ CV(κ1).
Corollary 5. If L ∈ ST and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ ST (κ1) if

|ℶ1ℶ2| (|ℶ1|+ 1) (|ℶ2|+ 1)

ℶ3(ℶ3+1)
G2,1(|ℶ1|+ 2, |ℶ2|+ 2;ℶ3 + 2, 1)

+
(3− κ1) |ℶ1ℶ2|

ℶ3
G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1, 1)

+ (1−κ1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3, 1)− 1] ≤ 1− κ1.

Corollary 6. IIf L ∈ CV and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R, then HOℶ1,ℶ2,ℶ3(L) ∈ ST (κ1) if

|ℶ1ℶ2|
ℶ3

G2,1(|ℶ1|+ 1, |ℶ2|+ 1;ℶ3 + 1, 1)

+ (1−κ1) [G2,1(|ℶ1| , |ℶ2| ;ℶ3, 1)− 1] ≤ 1− κ1.

Corollary 7. Let Υ1 given by (3) and ℶ1,ℶ2 ∈ C−{0}, ℶ3 ∈ R. If L ∈ ξ−ST for some
ξ(0 ≤ ξ < ∞) and satisfies

|ℶ1ℶ2|Υ1

ℶ3(1)1
G3,2(|ℶ1|+ 1, |ℶ2|+ 1,Υ1 + 1;ℶ3 + 1, 2; 1)

+ (1−κ1) [G3,2(|ℶ1| , |ℶ2| ,Υ1;ℶ3, 1; 1)− 1] ≤ 1− κ1,

then HOℶ1,ℶ2,ℶ3(L) ∈ ST (κ1).



REFERENCES 3397

4. Conclusions

Using of the Hohlov operator HOℶ1,ℶ2,ℶ3(L), we find necessary condition for this oper-
ator to be in the subfamilies Cκ3(κ1, κ2) and S∗κ3(κ1, κ2) of analytic functions with negative
coefficients. Furthermore, we investigate several inclusion properties for these subfamilies.
Also, our results will imply a number of corollaries. Hohlov operator HOℶ1,ℶ2,ℶ3(L) can
be used to derive new necessary and sufficient condition for analytic functions in various
subfamilies in the open unit disk.

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